The maximality conjecture for Gaussian Riemann derivatives

Let $n$ be a positive integer, and let $D_{\mathcal{A}}$ be an $n$-th generalized Riemann differentiation without excess. A generalized Riemann differentiation is called Gaussian when it is one of the Gaussian Riemann derivatives introduced in the paper, and symmetric Gaussian when it is the corresponding symmetric derivative. For a function $f$ and point $x$, write $f_{(k)}(x)$ and $f_{(k)}^s(x)$ for the $k$-th Peano and symmetric Peano derivatives, respectively.

Maximality conjecture for Gaussian Riemann derivatives. For all functions $f$ and points $x$:

1. If $D_{\mathcal{A}}$ is not Gaussian, then

$$\text{both }f_{(n-1)}(x)\text{ and }D_{\mathcal{A}}f(x)\text{ exist}\ \Longrightarrow\ f_{(n)}(x)\text{ exists}$$

should be false.

2. If $n\geq 3$ and $D_{\mathcal{A}}$ is symmetric but not symmetric Gaussian, then

$$\text{both }f_{(n-2)}^s(x)\text{ and }D_{\mathcal{A}}f(x)\text{ exist}\ \Longrightarrow\ f_{(n)}^s(x)\text{ exists}$$

should be false.

The conjecture identifies the Gaussian and symmetric Gaussian derivatives as the largest classes for which the generalized Marcinkiewicz–Zygmund equivalences can hold. Part (i) is known for $n=1,2$ and remains open for $n\geq 3$; part (ii) is false for $n=1,2$, true for $n=3,4$, and remains open for $n\geq 5$.